A385922 E.g.f. A(x) satisfies A(x) = exp(x*A(x) + x^5*A''''(x)).
1, 1, 3, 16, 125, 16296, 11929927, 30230776864, 203634850471929, 3082625458810336000, 93280255561776693446891, 5173509703646410927969711104, 491814532626655136406839912703157, 75968624000349445912469318939348786176, 18252829396078618393615717880609268502659375
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Keywords
Programs
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Mathematica
terms = 15; A[] = 1; Do[A[x] = Exp[x*A[x]+x^5*A''''[x]] + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] * Range[0,terms-1]! (* Stefano Spezia, Aug 04 2025 *)
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PARI
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=0, i-1, (1+j)*(1+sum(k=1, 4, stirling(4, k, 1)*j^k))*binomial(i-1, j)*v[j+1]*v[i-j])); v;
Formula
a(0) = 1; a(n) = Sum_{k=0..n-1} (1 + k) * (1 - 6*k + 11*k^2 - 6*k^3 + k^4) * binomial(n-1,k) * a(k) * a(n-1-k).